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<h1 class="heading"><a href="MATH-2023-OPDE.html"><span class="title">MATH 2023: Ordinary and Partial Differential Equations</span></a></h1>
<p class="byline">Xiaoyi Chen and Wei Zhang</p>
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<a href="ch_first.html" data-scroll="ch_first" class="internal"><span class="codenumber">1</span> <span class="title">Introduction</span></a><ul>
<li><a href="sec_1-intro.html" data-scroll="sec_1-intro" class="internal">Classification of Differential Equations</a></li>
<li><a href="sec_2-intro.html" data-scroll="sec_2-intro" class="internal">Linear and Nonlinear Equation</a></li>
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<a href="ch_second.html" data-scroll="ch_second" class="internal"><span class="codenumber">2</span> <span class="title">First Order Ordinary Differential Equations</span></a><ul>
<li><a href="sec2_1.html" data-scroll="sec2_1" class="internal">Linear Equations</a></li>
<li><a href="sec2_2.html" data-scroll="sec2_2" class="internal">Further Discussion of Linear Equations (For reading only)</a></li>
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<a href="ch_third.html" data-scroll="ch_third" class="internal"><span class="codenumber">3</span> <span class="title">third Order Linear Equations</span></a><ul>
<li><a href="sec3_1.html" data-scroll="sec3_1" class="internal">Homogeneous equations with constant coefficient</a></li>
<li><a href="sec3_2.html" data-scroll="sec3_2" class="internal">Fundamental Solutions of Linear Homogeneous Equations</a></li>
<li><a href="sec3_3.html" data-scroll="sec3_3" class="internal">Linear Independence and Wronskian</a></li>
<li><a href="sec3_4.html" data-scroll="sec3_4" class="internal">Complex roots of the characteristic equations</a></li>
<li><a href="sec3_5.html" data-scroll="sec3_5" class="internal">Repeated Roots: Reduction of Order</a></li>
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<a href="ch_four.html" data-scroll="ch_four" class="internal"><span class="codenumber">4</span> <span class="title">Higher Order Linear Equations</span></a><ul>
<li><a href="sec4_1.html" data-scroll="sec4_1" class="internal">General Theory of the <span class="process-math">\(n\)</span>-th Order Linear Equations</a></li>
<li><a href="sec4_2.html" data-scroll="sec4_2" class="internal">Homogeneous Equations with Constant Coefficients</a></li>
<li><a href="sec4_3.html" data-scroll="sec4_3" class="internal">The Method of Undetermined Coefficients</a></li>
<li><a href="sec4_4.html" data-scroll="sec4_4" class="internal">The Method of Variation of Parameters</a></li>
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<a href="ch_five.html" data-scroll="ch_five" class="internal"><span class="codenumber">5</span> <span class="title">Series Solutions of Second Order Linear Equations</span></a><ul>
<li><a href="sec5_1.html" data-scroll="sec5_1" class="active">Brief Review on Power Series</a></li>
<li><a href="sec5_2.html" data-scroll="sec5_2" class="internal">Introduction</a></li>
<li><a href="sec5_3.html" data-scroll="sec5_3" class="internal">Series Solutions Near an Ordinary Point</a></li>
<li><a href="sec5_4.html" data-scroll="sec5_4" class="internal">Euler’s Equation</a></li>
<li><a href="sec5_5.html" data-scroll="sec5_5" class="internal">Series Solution near a Regular Singular Point</a></li>
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<a href="ch_six.html" data-scroll="ch_six" class="internal"><span class="codenumber">6</span> <span class="title">System of First Order Linear Equations</span></a><ul>
<li><a href="sec6_1.html" data-scroll="sec6_1" class="internal">Introduction <span class="process-math">\(\&amp;\)</span> Basic Theory</a></li>
<li><a href="sec6_2.html" data-scroll="sec6_2" class="internal">Homogeneous System with Constant Coefficients</a></li>
<li><a href="sec6_3.html" data-scroll="sec6_3" class="internal">Complex Eigenvalues</a></li>
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<li><a href="sec6_5.html" data-scroll="sec6_5" class="internal">Fundamental Matrices</a></li>
<li><a href="sec6_6.html" data-scroll="sec6_6" class="internal">Non-homogeneous linear systems</a></li>
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<a href="ch_seven.html" data-scroll="ch_seven" class="internal"><span class="codenumber">7</span> <span class="title">Partial Differential Equations</span></a><ul>
<li><a href="sec7_1.html" data-scroll="sec7_1" class="internal">Two-Point Boundary Value Problems</a></li>
<li><a href="sec7_2.html" data-scroll="sec7_2" class="internal">Eigenvalue Problems</a></li>
<li><a href="sec7_3.html" data-scroll="sec7_3" class="internal">Fourier Series</a></li>
<li><a href="sec7_4.html" data-scroll="sec7_4" class="internal">The Fourier Convergence Theorem</a></li>
<li><a href="sec7_5.html" data-scroll="sec7_5" class="internal">Even and Odd Functions</a></li>
<li><a href="sec7_6.html" data-scroll="sec7_6" class="internal">Introduction to Partial Differential Equations</a></li>
<li><a href="sec7_7.html" data-scroll="sec7_7" class="internal">1D Heat Equation; Solutions by Separation of Variable and Fourier Series</a></li>
<li><a href="sec7_8.html" data-scroll="sec7_8" class="internal">Other Heat Conduction Problems</a></li>
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<a href="ch_eight.html" data-scroll="ch_eight" class="internal"><span class="codenumber">8</span> <span class="title">Laplace transform</span></a><ul>
<li><a href="sec8_1.html" data-scroll="sec8_1" class="internal">What are Laplace Transforms, and Why?</a></li>
<li><a href="sec8_2.html" data-scroll="sec8_2" class="internal">Finding Laplace Transforms</a></li>
<li><a href="sec8_3.html" data-scroll="sec8_3" class="internal">Finding inverse transforms using partial fractions</a></li>
<li><a href="sec8_4.html" data-scroll="sec8_4" class="internal">Solving ODEs and ODE Systems</a></li>
<li><a href="sec8_5.html" data-scroll="sec8_5" class="internal">Step input and Impulse problems</a></li>
<li><a href="sec8_6.html" data-scroll="sec8_6" class="internal">Laplace transform for PDE (heat equation)</a></li>
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<a href="ch_features.html" data-scroll="ch_features" class="internal"><span class="codenumber">9</span> <span class="title">Examples of PreTeXt features</span></a><ul><li><a href="sec_features-blocks.html" data-scroll="sec_features-blocks" class="internal">Environments and Blocks</a></li></ul>
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<li class="link"><a href="solutions-1.html" data-scroll="solutions-1" class="internal"><span class="codenumber">A</span> <span class="title">Selected Hints</span></a></li>
<li class="link"><a href="solutions-2.html" data-scroll="solutions-2" class="internal"><span class="codenumber">B</span> <span class="title">Selected Solutions</span></a></li>
<li class="link"><a href="appendix-1.html" data-scroll="appendix-1" class="internal"><span class="codenumber">C</span> <span class="title">List of Symbols</span></a></li>
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<main class="main"><div id="content" class="pretext-content"><section class="section" id="sec5_1"><h2 class="heading hide-type">
<span class="type">Section</span> <span class="codenumber">5.1</span> <span class="title">Brief Review on Power Series</span>
</h2>
<p id="p-175">Recall from calculus that a power series in <span class="process-math">\(x-x_0\)</span> is an infinite series of the form</p>
<div class="displaymath process-math" data-contains-math-knowls="">
\begin{equation*}
\sum_{n=0}^{\infty} a_n(x - x_0)^n=a_0 + a_1(x-x_0)+a_2(x-x_0)^2+\cdots
\end{equation*}
</div>
<p class="continuation">Such a series is also said to be a <dfn class="terminology">power series centered at <span class="process-math">\(x_0\)</span></dfn>. For example, the power series <span class="process-math">\(\sum_{n=0}^{\infty}(x+1)^n\)</span> is centered at <span class="process-math">\(x_0=-1\text{.}\)</span>The following list summarizes some important facts about power series.</p>
<ul id="p-176" class="disc">
<li id="li-4">
<p id="p-177"><dfn class="terminology">Convergence</dfn>  A power series <span class="process-math">\(\displaystyle\sum_{n=0}^{\infty} a_n(x - x_0)^n\)</span> <dfn class="terminology">converges</dfn> at a point <span class="process-math">\(x\)</span> if</p>
<div class="displaymath process-math" data-contains-math-knowls="">
\begin{equation*}
\lim_{m\to\infty}\sum_{n=0}^m a_n(x - x_0)^n
\end{equation*}
</div>
<p class="continuation">exists for that <span class="process-math">\(x\text{.}\)</span>The series certainly converges for <span class="process-math">\(x = x_0\text{;}\)</span> it may converge for all <span class="process-math">\(x\text{,}\)</span> or it may converge for some <span class="process-math">\(x\)</span> and not for others.</p>
</li>
<li id="li-5">
<p id="p-178"><dfn class="terminology">Absolute Convergence</dfn>  A power series <span class="process-math">\(\displaystyle\sum_{n=0}^{\infty} a_n(x - x_0)^n\)</span> <dfn class="terminology">converges absolutely</dfn> at a point <span class="process-math">\(x\)</span> if</p>
<div class="displaymath process-math" data-contains-math-knowls="">
\begin{equation*}
\sum_{n=0}^{\infty} |a_n(x - x_0)^n|=\sum_{n=0}^{\infty}|a_n||x-x_0|^n
\end{equation*}
</div>
<p class="continuation">converges.<em class="emphasis">If the series converges absolutely, then the series also converges</em>; however, the converse is not necessarily true.</p>
</li>
<li id="li-6">
<p id="p-179"><dfn class="terminology">Ratio Test</dfn>   Convergence of a power series can often be determined by the ratio test. If <span class="process-math">\(a_n\neq 0\text{,}\)</span> and if for a fixed value of <span class="process-math">\(x\)</span></p>
<div class="displaymath process-math" data-contains-math-knowls="">
\begin{equation*}
\textcolor{black}{\lim_{n\to\infty} \left|\frac{a_{n+1}(x - x_0)^{n+1}}{a_n(x - x_0)^n}\right|}=|x-x_0|\lim_{n\to\infty} \left|\frac{a_{n+1}}{a_n}\right|\textcolor{black}{=L|x-x_0|.}
\end{equation*}
</div>
<p class="continuation">The power series converges absolutely at that value of <span class="process-math">\(x\)</span> if <span class="process-math">\(|x-x_0|&lt;1/L\text{,}\)</span> and diverges if <span class="process-math">\(|x-x_0|&gt;1/L\text{.}\)</span> If <span class="process-math">\(|x-x_0|=1/L\text{,}\)</span> the test is inconclusive.</p>
</li>
<li id="li-7">
<p id="p-180">The <dfn class="terminology">radius of convergence</dfn> (about <span class="process-math">\(x_0\)</span>): a nonnegative <span class="process-math">\(\textcolor{blue}{\rho}\text{,}\)</span> such that</p>
<div class="displaymath process-math" data-contains-math-knowls="">
\begin{equation*}
\sum_{n=0}^{\infty} a_n(x-x_0)^n ~~\left\{\begin{array}{ll} \text{\textcolor{blue}{converges absolutely}} &amp; \text{for $\textcolor{magenta}{|x-x_0|&lt;}\textcolor{blue}{\rho}$},\\ \text{diverges} &amp;\text{for $|x-x_0|&gt;\textcolor{blue}{\rho}$.}\end{array}\right.
\end{equation*}
</div>
<p id="p-181">For a series that converges only at <span class="process-math">\(x_0\text{,}\)</span> <span class="process-math">\(\rho=0\text{;}\)</span>for a series that converges for all <span class="process-math">\(x\text{,}\)</span> <span class="process-math">\(\rho\)</span> is infinite.</p>
<p id="p-182">If <span class="process-math">\(\rho&gt;0\text{,}\)</span> then the interval <span class="process-math">\(|x-x_0|&lt;\rho\)</span> is called <dfn class="terminology">the interval of convergence</dfn>.</p>
</li>
<li id="li-8">
<p id="p-183"><dfn class="terminology">A Power Series Defines a Function</dfn>  </p>
<div class="displaymath process-math" data-contains-math-knowls="">
\begin{equation*}
f(x)=\displaystyle\sum_{n=0}^{\infty} a_n(x-x_0)^n
\end{equation*}
</div>
<p class="continuation">whose domain is the interval of convergence of the series.</p>
<ul id="p-184" class="circle">
<li id="li-9">
<p id="p-185">Within the interval of convergence, <span class="process-math">\(f'(x)\)</span> and <span class="process-math">\(\int f(x)\textrm{d}x\)</span> can be found by term-wise differentiation and integration. For example,</p>
<div class="displaymath process-math" data-contains-math-knowls="">
\begin{equation*}
f'(x)=\sum_{n=0}^{\infty} na_n(x-x_0)^{n-1}=\sum_{\textcolor{blue}{n=1}}^{\infty} na_n(x-x_0)^{n-1},
\end{equation*}
</div>
<div class="displaymath process-math" data-contains-math-knowls="">
\begin{equation*}
f''(x)=\sum_{n=0}^{\infty} n(n-1)a_n(x-x_0)^{n-2}=\sum_{\textcolor{blue}{n=2}}^{\infty} n(n-1)a_n(x-x_0)^{n-2}
\end{equation*}
</div>
</li>
<li id="li-10"><p id="p-186">The differentiated (integrated) series has <em class="emphasis">the same radius of convergence</em> as the original series.</p></li>
<li id="li-11"><p id="p-187">However, <em class="emphasis">convergence at an endpoint</em> may be either lost by differentiation or gained through integration.</p></li>
</ul>
<p id="p-188">These results are important and will be used shortly.</p>
</li>
<li id="li-12">
<p id="p-189"><dfn class="terminology">Useful Expansions</dfn>   Some power series expansions about 0 that you should be familiar with.Except for the last one, the expansions are valid for all <span class="process-math">\(z\)</span> (i.e. the radius of convergence is <span class="process-math">\(\rho=\infty\)</span>).</p>
<div class="displaymath process-math" data-contains-math-knowls="">
\begin{equation*}
e^z=\sum_{n=0}^{\infty}\frac{z^n}{n!}
\end{equation*}
</div>
<div class="displaymath process-math" data-contains-math-knowls="">
\begin{equation*}
\cos(z)=\sum_{k=0}^{\infty}(-1)^k\frac{z^{2k}}{(2k)!},\quad \sin(z)=\sum_{k=0}^{\infty}(-1)^k\frac{z^{2k+1}}{(2k+1)!}
\end{equation*}
</div>
<div class="displaymath process-math" data-contains-math-knowls="">
\begin{equation*}
\textcolor{red}{\frac{1}{1-z}=\sum_{n=0}^{\infty}z^n}~~~~(|z|&lt;1)
\end{equation*}
</div>
</li>
<li id="li-13">
<p id="p-190"><dfn class="terminology">Identity Property</dfn>  </p>
<ul id="p-191" class="circle">
<li id="li-14">
<p id="p-192">If <span class="process-math">\(\displaystyle\sum_{n=0}^{\infty} c_n(x-x_0)^n=0\)</span> for all <span class="process-math">\(x\)</span> in the interval of convergence, then</p>
<div class="displaymath process-math" data-contains-math-knowls="">
\begin{equation*}
c_n=0\quad \text{ for all } n.
\end{equation*}
</div>
</li>
<li id="li-15">
<p id="p-193">If <span class="process-math">\(\displaystyle\sum_{n=0}^{\infty} a_n(x-x_0)^n=\sum_{n=0}^{\infty} b_n(x-x_0)^n\)</span> for all <span class="process-math">\(x\)</span> in the interval of convergence, then</p>
<div class="displaymath process-math" data-contains-math-knowls="">
\begin{equation*}
\textcolor{magenta}{a_n=b_n\quad \text{ for all } n}.
\end{equation*}
</div>
<p id="p-194">Namely, if two power series are <dfn class="terminology">equal</dfn>, then they must have the same coefficients.</p>
</li>
</ul>
</li>
<li id="li-16">
<p id="p-195"><dfn class="terminology">Arithmetic of Power Series</dfn>   </p>
<div class="displaymath process-math" data-contains-math-knowls="">
\begin{equation*}
f(x)=\displaystyle\sum_{n=0}^{\infty} a_n(x-x_0)^n,\qquad g(x)=\displaystyle\sum_{n=0}^{\infty} b_n(x-x_0)^n
\end{equation*}
</div>
<p class="continuation">whose domain is the interval of convergence of the series.</p>
<ul id="p-196" class="circle">
<li id="li-17"><div class="displaymath process-math" data-contains-math-knowls="" id="p-197">
\begin{equation*}
f(x)\pm g(x)=\sum_{n=0}^{\infty} (a_n\pm b_n)~(x-x_0)^n=\sum_{n=0}^{\infty} c_n(x-x_0)^n.
\end{equation*}
</div></li>
<li id="li-18">
<div class="displaymath process-math" data-contains-math-knowls="" id="p-198">
\begin{equation*}
f(x)g(x)=\left[\sum_{n=0}^{\infty} a_n(x-x_0)^n\right]\left[\sum_{n=0}^{\infty} b_n(x-x_0)^n\right]=\sum_{n=0}^{\infty} d_n(x-x_0)^n,
\end{equation*}
</div>
<p class="continuation">where <span class="process-math">\(d_n=a_0b_n+a_1b_{n-1}+\cdots+a_nb_0\text{.}\)</span></p>
</li>
<li id="li-19"><p id="p-199">If <span class="process-math">\(g(x_0)\neq0\text{,}\)</span> the series for <span class="process-math">\(f(x)/g(x)\)</span> can also be found accordingly.</p></li>
</ul>
</li>
<li id="li-20">
<p id="p-200"><dfn class="terminology">Analytic at a Point</dfn>   A function <span class="process-math">\(f\)</span> is <dfn class="terminology">analytic at a point <span class="process-math">\(x_0\)</span></dfn> if it can be represented by a power series in <span class="process-math">\(x-x_0\)</span> with a positive or infinite radius of convergence (i.e. <span class="process-math">\(\rho&gt;0\)</span>).In calculus, functions like <span class="process-math">\(e^x\text{,}\)</span> <span class="process-math">\(\cos(x)\text{,}\)</span> <span class="process-math">\(\sin(x)\)</span> can be represented by Taylor series with <span class="process-math">\(|x|&lt;\infty\text{.}\)</span></p>
<p id="p-201">Taylor series centered at 0 are called <dfn class="terminology">Maclaurin series</dfn>. Show that <span class="process-math">\(e^x\text{,}\)</span> <span class="process-math">\(\cos(x)\text{,}\)</span> <span class="process-math">\(\sin(x)\text{,}\)</span> are analytic at <span class="process-math">\(x=0\text{.}\)</span></p>
</li>
<li id="li-21"><p id="p-202"><dfn class="terminology">Shifting the Summation Index</dfn></p></li>
</ul></section></div></main>
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